Analytical study of one dimensional time fractional Schrödinger problems arising in quantum mechanics

This work presents the analytical study of one dimensional time-fractional nonlinear Schrödinger equation arising in quantum mechanics. In present research, we establish an idea of the Sumudu transform residual power series method (ST-RPSM) to generate the numerical solution of nonlinear Schrödinger models with the fractional derivatives. The proposed idea is the composition of Sumudu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken in Caputo sense. The proposed technique is unique since it requires no assumptions or variable constraints. The ST-RPSM obtains its results through a series of successive iterations, and the resulting form rapidly converges to the exact solution. The results obtained via ST-RPSM show that this scheme is authentic, effective, and simple for nonlinear fractional models. Some graphical structures are displayed at different levels of fractional orders using Mathematica Software.


Definition 2.1
The Riemann-Liouville of order α > 0 for expression of fractional integral operator is 36 Definition 2. 2 The fractional derivative of S(ℵ, ℘) in Caputo form is 36 .Definition 2. 3 Let a series such as 37 ( www.nature.com/scientificreports/which known as the power series in fractional form and ℘ = ℘ 0 , where ℘ is a a parameter and ζ m are set values in the solution of series function. Theorem 2.1 37 Consider ζ shows a power series in fractional form when ℘ = ℘ 0 in terms of where 0 < n − 1 < α ≤ n, ℵ ∈ I, ℘ 0 ≤ ℘ < ℘ 0 + R .Proof Let S(ℵ, ℘) be a function of ℵ and ℘ , representing the power series of Eq. ( 3) in multiple fractional form.If we put ℘ = ℘ 0 in Eq. ( 4), then only its first component remains, and other components will vanish, so we obtain Applying the D α ℘ operator to Eq. ( 4), the subsequent expansion yields On substituting ℘ = ℘ 0 to Eq. ( 7), we determine the value of ζ 1 (ℵ) such as Now, using an operator of D α ℘ one time on Eq. ( 7), the following expansion takes place: On substituting ℘ = ℘ 0 to Eq. ( 9), we determine the value of ζ 2 (ℵ) such as By repeating the operator D α ℘ m times and then using ℘ = ℘ 0 , we can obtain the sequence of ζ m (ℵ) such as: which shows the agreement with Eq. ( 9).This proves the theorem.
Note By applying the series of ζ m (ℵ) from Eq. (11) into Eq.( 4), it is possible to obtain the power series of S(ℵ, ℘) in multiple fractional form at the value of ℘ = ℘ 0 as follows, that represents the algorithm of Taylor's formula in a generalized form.Now, if α = 1 , Eq. ( 12) turns to Taylor's series formula in a classical form such as Thus, a new generalization form of Eq. ( 12) has been derived which helps to obtain the outcomes of timefractional NLSE.

Definition 2.4
The Sumudu transform on the set of functions A is expressed as 38 : where M is constant for a finite parameter of a function in the set A, and k 1 , k 2 may or may not be finite.Now, the integral relation of Sumudu transform is expressed as www.nature.com/scientificreports/ The subsequent aspects of Sumudu transform are expressed as 39 :

Definition 2.5
The fractional order of ST in Caputo sense is defined as 35

Formulation of ST-RPSM
The present segment describes the formulation of ST-RPSM for the computational analysis of time-fractional NLSE.We observe that ST has the capability of changing fractional orders to a traditional space.This relation can easily be handled by RPSM to a system of algebraic equations which is close to the precise results of the fractional problems.We this formulation by a few steps where the complete procedure is described.Let S(ℵ, ℘) and ̺(ℵ) be two complex functions with real and imaginary parts such that where ϑ(ℵ, ℘) and ψ(ℵ, ℘) be two real valued functions over ℵ ∈ R , ζ(ℵ) and η(ℵ) are analytic functions on ℵ ∈ R .By using Eq. ( 16), the system of Eq. ( 1) can be transformed to the following PDEs system such that whereas the conditions becomes as Now, in general, the solution of Eq. ( 17) along condition (18) presents the solution of Eq. (1) with conditions (2).Hence, we develop the idea of ST-RPSM for Eq. ( 17) along condition (18).We will explain this concept with the following steps: Step 1 In our first stage, we execute the Sumudu transform to Eq. ( 17) and converting it to another space using condition (18), we get where P(ℵ, θ) = S{ϑ(ℵ, ℘)} and R(ℵ, θ) = S[ψ(ℵ, ℘)].
Step 2 In the second stage, Let the precise results of Eq. ( 19) for P(ℵ, θ) and R(ℵ, θ) have the subsequent expansions and the k-th truncated Sumudu series of Eq. ( 20) is expressed as Vol One can determine the constants ζ n and η n of the expansion series in Eq. ( 21) by calculating the k-th Sumudu residual functions.
Step 3 At the third stage, we develop the residual formula, such that Res 1 and Res 2 show the iterative formula for Eq. ( 19) as follows Hence, the k-th truncated Sumudu residual series for Eq. ( 22) becomes as There are some significant results from the RPSM such as Step 4 Put the k-th truncated Sumudu series of Eq. ( 21) to the k− th Sumudu residual function of Eq. ( 23).
Step 5.The coefficients of ζ k (ℵ) and η k (ℵ) can be derived by using lim The derived coefficients are compiled in terms of an iterative series for P k (ℵ, θ) and R k (ℵ, θ) of the expansion (21).
Step 6 By utilizing inverse ST to the obtained resultant series, we can achieve the approximate solution P k (ℵ, θ) and R k (ℵ, θ) of the main fractional problem (17).

Numerical applications
In this segment, we show thevalidity, performance, and power of ST-RPSM by considering three applications of time fractional NLSE with different conditions.The Mathematica package has been utilized for performing complex conceptualization and computations involving mathematics.

Problem 1
Consider one dimensional NLSE of time-fractional in the subsequent form 21,40 in relates with following conditions The Eq. ( 24) can be simply transformed into the following fractional system: whereas the conditions becomes as By applying ST to Eq. ( 26) along condition (27) and simplifying it, we get Consider the precise result of Eq. ( 26) in the k-th expression such as Now, the k-th expression of Eq. ( 28) is also formed as Vol:.( 1234567890) www.nature.com/scientificreports/ The 1st coefficient of Eq. ( 29) can be derived by utilizing the following truncated succession of first order By using using Eq. ( 31) into Eq.( 30) at k = 1 , we obtain 1st ST-RPSM result Now, using the facts of RPSM such as limit of θ → ∞ for Eq. ( 32), we derive the results such as Similarly, the 2nd coefficient of Eq. ( 29) can be derived by utilizing the following truncated succession of second order By using using Eq. ( 34) into Eq.( 30) at k = 2 , we obtain 2nd ST-RPSM result Now, using the facts of RPSM such as limit of θ → ∞ for Eq. ( 35), we derive the results such as On similar way, we can derive the ST-RPSM results for k = 3, 4 Hence, the 4th ST-RPSM results of Eq. ( 29) can be expressed in the following series such as By applying inverse ST on system of Eq. ( 38), we obtain By using the similar process, this series can be continuous and thus ST-RPSM results for ϑ(ℵ, ℘) and ψ(ℵ, ℘) can converge to the following form whereas Considering α = 1 in Eqs. ( 40) and ( 24) has the subsequent precise solution (30)    P 4 (ℵ, θ) = cos 3ℵ − 9 sin 3ℵθ α − 81 cos 3xs 2α + 729 sin 3xs 3α + 6561 cos 3xs 4α , R 4 (ℵ, θ) = sin 3ℵ + 9 cos 3ℵθ α − 81 sin 3xs 2α − 729 cos 3xs 3α + 6561 sin 3xs 4α . ( (40) S(ℵ, ℘) = e 3iℵ χ(℘).www.nature.com/scientificreports/which has similar results with decomposition approach 22 , homotopy analysis method 41 , and variational approach 42 .Therefore, we can demonstrate that ST-RPSM is an easy, fundamental, and effective technique for solving fractional problems.Figure 1 demonstrates the graphical representation of time fractional NLSE in the shape of real and imaginary visuals and we divide it into its four graphical structures.Figure 1a shows the physical behavior of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.5 with −1 ≤ ℵ ≤ 1, 0 ≤ ℘ ≤ 0.5 .Figure 1b shows the physical description of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.8 with −1 ≤ ℵ ≤ 1, 0 ≤ ℘ ≤ 1 .Figure 1c illustrates the physical explana- tion of ST-RPSM outcomes in the form of 3D graphical structure by taking the fractional order at α = 1 with 0 ≤ ℵ ≤ 0.5, 0 ≤ ℘ ≤ 0.05 .Figure 1d illustrates the physical assessment of precise results for NLSE in the form of 3D graphical structure at 0 ≤ ℵ ≤ 0.5, 0 ≤ ℘ ≤ 0.05 .From these graphical structures, one can observe that time-fractional NSE has the full agreement of precise results with ST-RPSM outcomes at α = 1.

Problem 2
Consider one dimensional NLSE of time-fractional in the subsequent form 21,40 in relates with following conditions The Eq. ( 42) can be simply transformed into the following fractional system: whereas the conditions becomes as Now, the k-th expression of Eq. ( 46) is also formed as Using the ST-RPSM strategy, we can derive the results for k = 1, 2, 3, 4 such that Hence, the 4th ST-RPSM results of Eq. ( 47) can be expressed in the following series such as By applying inverse ST on system of Eq. (50), we obtain By using the similar process, this series can be continuous and thus ST-RPSM results for ϑ(ℵ, ℘) and ψ(ℵ, ℘) can converge to the following form whereas Considering α = 1 in Eq. (52), the system of Eqs. ( 24) along ( 25) has the subsequent precise result (46) � . (49) S(ℵ, ℘) = e iℵ χ(℘).www.nature.com/scientificreports/which has similar results with decomposition approach 22 , homotopy analysis method 41 , and variational approach 42 .Therefore, we can demonstrate that ST-RPSM is an easy, fundamental, and effective technique for solving fractional problems.Figure 2 demonstrates the graphical representation of time fractional NLSE in the shape of real and imaginary visuals and we divide it into its four graphical structures.Figure 2a shows the physical behavior of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.5 with 0 ≤ ℵ ≤ 3, 0 ≤ ℘ ≤ 0.5 .Figure 2b shows the physical description of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.8 with −3 ≤ ℵ ≤ 3, 0 ≤ ℘ ≤ 1 .Figure 2c illustrates the physical explana- tion of ST-RPSM outcomes in the form of 3D graphical structure by taking the fractional order at α = 1 with −5 ≤ ℵ ≤ 5, 0 ≤ ℘ ≤ 0.5 .Figure 2d illustrates the physical assessment of precise results for NLSE in the form of 3D graphical structure at α = 1 with −5 ≤ ℵ ≤ 5, 0 ≤ ℘ ≤ 0.5 .From these graphical structures, one can observe that time-fractional NSE has the full agreement of precise results with ST-RPSM outcomes at α = 1.

Problem 3
Consider one dimensional NLSE of time-fractional in the subsequent form 21,40 in relates with following conditions The Eq. ( 54) can be simply transformed into the following fractional system: whereas the conditions becomes as Now, the k-th expression of Eq. ( 58) is also formed as Using the ST-RPSM strategy, we can derive the results for k = 1, 2, 3, 4 such that Hence, the 4th ST-RPSM results of Eq. ( 59) can be expressed in the following series such as By applying inverse ST on system of Eq. (62), we obtain By using the similar process, this series can be continuous and thus ST-RPSM results for ϑ(ℵ, ℘) and ψ(ℵ, ℘) can converge to the following form Considering α = 1 in Eq. (64), Eq. ( 54) has the subsequent precise solution (57) ϑ(ℵ, 0) = 6sech 2 (4ℵ) 1 4 , ψ(ℵ, 0) = 0. ( � . (61) , S(ℵ, ℘) = 6 sech 2 (4ℵ) www.nature.com/scientificreports/which has similar results with decomposition approach 22 , homotopy analysis method 41 , and variational approach 42 .Therefore, we can demonstrate that ST-RPSM is an easy, fundamental, and effective technique for solving fractional problems.Figure 3 demonstrates the graphical representation of time fractional NLSE in the shape of real and imaginary visuals and we divide it into its four graphical structures.Figure 3a shows the physical behavior of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.5 with 0 ≤ ℵ ≤ 2, 0 ≤ ℘ ≤ 0.5 .Figure 3b shows the physical description of ST-RPSM solution in the form of 3D graphical structure by taking the fractional order at α = 0.8 with −1 ≤ ℵ ≤ 1, 0 ≤ ℘ ≤ 0.5 .Figure 3c illustrates the physical explana- tion of ST-RPSM outcomes in the form of 3D graphical structure by taking the fractional order at α = 1 with −1 ≤ ℵ ≤ 1, 0 ≤ ℘ ≤ 0.5 .Figure 3d illustrates the physical assessment of precise results for NLSE in the form of 3D graphical structure at α = 1 with −1 ≤ ℵ ≤ 1, 0 ≤ ℘ ≤ 0.5 .From these graphical structures, one can observe that time-fractional NLSE has the full agreement of precise results with ST-RPSM outcomes when α = 1.

Conclusion
In this work, we successfully obtained the approximate solutions of numerical results of time-fractional NLSE by utilizing the composition of Sumudu transform and residual power series scheme.The suggested approach yields the series results in a convergence form with minimal computational iterations.The Sundum transform can transfer the fractional order into a recurrence relation and the residual power series scheme can easily derive the series results from an algebraic system of fractional equations.The residual power series scheme has an excellent ability to handle nonlinear problems in time fractional models.We demonstrated the effectiveness and dependability of the suggested approach with three nonlinear models under Caputo fractional derivatives.A graphic illustration is used for analyzing the calculated results.The solutions provided to the time-fractional NLSE indicate the wave function of a quantum framework.Numerical computations are produced to show the graphical depictions of these wave functions, which offer valuable insights into the spatial dispersion of the quantum particle as it expands over time.These visualizations aid in comprehending the concept of the duality of wave particles and the uncertain characteristics of quantum physics.The derived results show that the obtained series results for different levels of fractional order confirm the reliability, comparability, and simplicity of the If D mα ℘ S(ℵ, ℘) are continuous on I × (℘ 0 , ℘ 0 + R), m = 0, 1, 2, • • • , thus values for ζ m (ℵ) are in which D mα ℘ = D α ℘ .D α ℘ .• • • D α ℘ (n − times).